Q.Derive an expression for the elastic energy stored per unit volume of a wire.

When a wire is stretched, work is done against the internal restoring force developed in it, and this work is stored in the wire as ELASTIC POTENTIAL ENERGY. Consider a wire of un-stretched length L and cross-sectional area A, stretched (within the elastic limit, with no energy lost) by a force that produces a final extension l. The small increment of work done in stretching the wire a further dl is $dW=F\,dl$, so the TOTAL work done in stretching it from zero extension to l is $W=\int_0^l F\,dl$. Since Young's modulus gives $Y=\dfrac{F/A}{l/L}$, the force at any intermediate extension $l'$ is $F=\dfrac{YAl'}{L}$; substituting and integrating, $W=\int_0^l\dfrac{YAl'}{L}dl'=\dfrac{YAl^2}{2L}=\dfrac12\left(\dfrac{YAl}{L}\right)l=\dfrac12 Fl$. So the ELASTIC POTENTIAL ENERGY stored in a stretched wire is $W=\tfrac12 Fl$ -- exactly half the product of the final force and the final extension (because the force grew linearly from zero up to F as the wire stretched, by Hooke's law). Dividing by the wire's volume $AL$ gives the ENERGY DENSITY (elastic energy stored per unit volume): $u=\dfrac{W}{AL}=\dfrac12\left(\dfrac{F}{A}\right)\left(\dfrac{l}{L}\right)=\dfrac12\times\text{stress}\times\text{strain}$; since stress equals Young's modulus times strain (Hooke's law), this can equally be written $u=\dfrac12 Y(\text{strain})^2$ -- a form that is especially convenient when only the strain (not the raw force) is known or asked for. This elastic energy is what is released as kinetic energy, sound, or heat when a stretched material (a bowstring, a catapult's elastic band, a compressed spring) is suddenly released.

Queries like "elastic potential energy in a stretched wire formula" and "energy density formula physics class 11" recur around the Mechanical Properties of Solids chapter of the NCERT/CBSE Class 11 Physics curriculum, a section tested regularly in board exams and JEE Main. The derivation shown here, $W=\tfrac12Fl$ and $u=\tfrac12\times\text{stress}\times\text{strain}$, is exactly the kind of first-principles proof competitive exams like to ask for rather than just the final formula.